Birkhoff subfibrations of the codomain fibration

A. S. Cigoli and S. Mantovani

Slice categories of a semi-abelian category C have a regular epireflection to their subcategories of internal Mal'tsev algebras. These are Birkhoff reflections, hence admissible with respect to regular epis in the sense of Janelidze's categorical Galois theory. We prove that when C is moreover peri-abelian, these reflections form an admissible Galois structure for a larger class of morphisms, called proquotients. Starting from a careful investigation of the previous situation, we prove that all regular epireflective subfibrations in Fib(C) of the codomain fibration of C can be constructed from a reflective subcategory M_0 of C whose unit morphisms have characteristic kernel. The fibres of such reflective subfibrations are admissible with respect to proquotients precisely when M_0 is a Birkhoff subcategory of C.

Keywords: categorical Galois theory; reflective subfibration; characteristic subobject; peri-abelian category

2020 MSC: 18D30; 18E13; 18E50

Theory and Applications of Categories, Vol. 40, 2024, No. 10, pp 301-323.

Published 2024-04-26.

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